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Tamurlan Tsarnaev, the elder of the two brothers accused of the Boston bombings, for example, was interviewed by the FBI in 2011. The bureau also has said agents trawled his online communications and other activity, but found nothing that warranted a full-scale investigation.

It was unclear Thursday how the two men accused of the Woolwich attack had come to the attention of MI5.

The British government said in a statement that broader powers to surveil online and other electronic communications would not have helped avert the attack.

“There are already substantial powers in place to track the communications of criminals and terrorists,” said a spokesman for Deputy Prime Minister Nick Clegg. “There is currently no suggestion” that government proposals in a draft wiretapping authority bill “would have had any relevance to yesterday’s sickening events.”

Prof. Errol G. Southers of UCLA, a former senior federal law enforcement official and one-time candidate to head the TSA, told The Times cases of homegrown violent extremism underline “the complex challenges associated with securing a democracy” against lone-wolf terrorism.